Legendre polynomials and Ramanujan-type series for 1/π

Abstract

We resolve a family of recently observed identities involving 1/π using the theory of modular forms and hypergeometric series. In particular, we resort to a formula of Brafman which relates a generating function of the Legendre polynomials to a product of two Gaussian hypergeometric functions. Using our methods, we also derive some new Ramanujan-type series.

A new series for π3and related congruences

“…Sun [33,39] 2 , In general, the corresponding p-adic congruences of these seven-type series involve linear combinations of two Legendre symbols. The author's conjectural series of types I-V and VII were studied in [6,48,53]. The author's three conjectural series of type VI and two series of type VII remain open.…”

Short Walk Adventures

“…where P k denotes the k-th Legendre polynomial, and the latter generating function is a particular instance of the Bailey-Brafman formula [15,34].…”

Holonomic Alchemy and Series for $$1/\pi $$

“…This expectation is shown to be true in many cases and it is the driving force behind the universal methods of establishing Sun's conjectures and similar identities in [6,11,17,21,22]. Two further examples of such factorizations follow from the two-variable identities The former transformation (9.2) allows one to deal with [19,Conjecture (3.29)] (namely, by making it equivalent to [19,Eq. (I3)] established in [6]), while the latter one (9.3) paves the ground for proving the family of conjectures (3.N ′ ) on Sun's list [19] in exactly the same way as in [17]. A drawback of using such two-variable factorizations in the proofs of the formulas for 1/π is the relatively cumbersome analysis: compare our proof of Sun's Conjecture (3.24 ′ ) from Section 8 with the proof of his (equivalent) Conjecture (3.24) given in [17].…”